I'd say not necessarily. Surely you can wear a tie to smarten things up a bit. However, those shirts look really good simply buttoned all the way up to the top (as seen in the picture you provided). Here are a few more pictures of tie-less buttoned men from some recent lookbooks:

Guy on the right:

I know you specified button-down collars, but I'd give the same suggestion for no button as well. I'd say it's largely personal opinion though. As for what kind of tie, a solid knitted tie works fairly well in many settings.

Agreed, but you could wear a slim tie with it. It can be hard to find nice ties slim enough for low collars, though.

For that ASOS shirt you used as an example, I'd wear a brown herringbone tweed tie. H&M used to have a good selection of slim ties. I bought one there, and it looks a bit like the image below, but slightly lighter in colour.